Instead of representing each query and document as a single fixed-length vector, multi-vector retrieval represents them with multiple token vectors, mainly to improve the limited capacity of fixed-length representations.
Specifically, a query \(Q=\lbrace q_1, \dots , q_n\rbrace \)  and a document \(D=\lbrace d_1, \dots , d_m\rbrace \)  are encoded into a set of vectors \(\lbrace {\mathbf {q}}_1, \dots , {\mathbf {q}}_n\rbrace \)  and \(\lbrace {\mathbf {d}}_1, \dots , {\mathbf {d}}_m\rbrace \) .
The similarity function between a query and a document is re-defined for multi-vector retrieval.
For instance, ColBERT [1]} designs the similarity function as follows:
\(\text{sim}(Q,D) = \sum _{i=1}^n \max _{j=1\dots m} {\mathbf {q}}_i^\top {\mathbf {d}}_j.\) 
